Forming Obtuse Triangles with Sides 8 cm, 15 cm, and x cm: An SEO-Optimized Guide

A right or obtuse triangle is a fundamental concept in geometry. In this article, we will explore how to determine the number of possible right and obtuse triangles that can be formed with integer side lengths 8 cm, 15 cm, and x cm.

Understanding the Problem

The challenge here is to find the integer values of x which can form an obtuse triangle with given sides 8 cm and 15 cm. This involves two major steps: confirming that the three sides can form a triangle and ensuring that the triangle is obtuse. Let's break it down.

Triangle Inequality Conditions

For three sides to form a triangle, they must satisfy the triangle inequalities:

8 15 x → x 23 8 x 15 → x 7 15 x 8 → x -7 (always true since x 7)

Combining these inequalities, we get:

7 x 23, which means x can take any integer value from 8 to 22. Thus, there are a total of 15 possible integer values for x.

Conditions for an Obtuse Triangle

To determine if a triangle is obtuse, we use the following condition: if the square of the longest side is greater than the sum of the squares of the other two sides, the triangle is obtuse. Let's now apply this to our problem. We will consider two cases: when x is the longest side and when 15 is the longest side.

Case 1: x is the Longest Side

For x to be the longest side, it must satisfy the inequality:

x2 82 152 → x2 289

Taking the square root of both sides, we get:

x 17

The possible integer values for x in this case are 18, 19, 20, 21, and 22, giving us 5 possible values.

Case 2: 15 is the Longest Side

For 15 to be the longest side, it must satisfy the inequality:

152 82 x2 → 225 64 x2

Rearranging, we get:

x2 161 → x √161

Since √161 ≈ 12.688, the possible integer values for x are 8, 9, 10, 11, and 12, giving us another 5 possible values.

Conclusion

Adding the valid values from both cases:

5 values when x is the longest side (18, 19, 20, 21, 22) 5 values when 15 is the longest side (8, 9, 10, 11, 12)

Thus, the total number of obtuse triangles that can be formed is 10. Therefore, there are 10 such obtuse triangles possible with sides 8 cm, 15 cm, and x cm.

Understanding Integer Side Lengths and Right Angles

If x were 23, the triangle would be a right triangle. Therefore, x must be between 18 and 22 to ensure the triangle is obtuse. We can visualize this with a simple graph or diagram to understand the distribution of possible integer values.

SEO and Keyword Optimizations

Keywords: obtuse triangle, forming triangles, integer side lengths

By optimizing content with these keywords, the article is more likely to be found by search engines and potential readers interested in geometry and triangle formation.

To reinforce the educational value of this content, consider the following resources:

A YouTube video that visually demonstrates the formation of triangles. A Python code snippet that calculates the number of possible triangles. A Biology and Mathematics Cross-Reference to highlight the interdisciplinary nature of these concepts.